The mysteries of spin

There’s a new post with this title over at Peter Woit’s blog, in which he “explains” spin in the traditional way using the group SU(2), as has been done consistently since the early 1920s. Well, when I say “consistently”, the problem is it isn’t consistent. But the inconsistency doesn’t become apparent until you’ve moved a long way from these fundamentals, so that the cause of the inconsistency is no longer obvious. What appears to me to be the problem is that this theory tries to explain spin without a concept of time. It is impossible to describe classically rotating objects without time, and I believe is it equally impossible to describe quantum spin without time.

The question then is, what sort of time do you need for quantum mechanics? Woit makes a big thing in other places about using Euclidean spacetime for quantum mechanics, rather than Minkowski spacetime, which is appropriate for relativity. Now it makes a big difference to the mathematics which you use. Philosophically, the question is, does an individual elementary particle have a concept of time? Philosophically, my answer is “no”. My reasoning is that in order to measure time, one needs a clock, and for a clock, one needs an atom.

For precise measurements of time on Earth, a caesium atom is ideal. It ticks so fast that by counting the ticks one can measure time extremely accurately. But a hydrogen atom is big enough – it is used to measure the age of stars, galaxies and the universe in general. However, once you pull apart the electron and the proton, and treat them in isolation, the clock is broken. It no longer ticks. At least, that is my view. Others believe that the electron still ticks even if nothing observes it. Others believe that the electron experiences a continuous time, But let’s not get involved in this philosophical argument. I shall show you that in either case, the standard mathematical description of spin is wrong. The correct descriptions are quite different in the two cases, but neither of them resembles the standard 1920s picture.

Let’s assume first of all that the electron ticks, but that it cannot remember how many times it has ticked. The group SO(3) does not tick. The group SO(3,1) cannot tick if time goes in one direction. The only four-dimensional group that ticks is SO(4). It ticks because it is a commuting product of two copies of SU(2), and the electron ticks like a pendulum clock, between the left-handed state and the right-handed state. The group that you need to describe an electron in isolation is therefore SO(4). Not Spin(4). Not Spin(3,1). Not SO(3,1).

Now let’s assume instead that the electron has a continuous concept of time. If it ticks, it can count the ticks. If it doesn’t tick, it feels time in some other way. Either way, it requires a copy of the real numbers (or the integers) in its symmetry group. Since we cannot measure the direction of spin, it only requires a group SO(2,1), not SO(3,1). But it must have the double cover of SO(2,1), that is SL(2,R), because it has spin 1/2. Now the group theory is the same as for a classically rotating particle, such as the Earth, if you distinguish day from night. Of course, you cannot say that the Earth is in the “day” state or the “night” state, because that depends on the observer. But any particular observer or experiment can distinguish the two states very clearly.

Likewise, I am not saying that the electron “is” a classically rotating particle, but the underlying group theory is the same. Now the important bit is what happens when an electron and a proton interact, and their two copies of SL(2,R) generate a bigger group? Dirac (1928) assumed (and I emphasise this, assumed) they generate SL(2,C). But is this correct? This is not how classically rotating particles interact, so why should we think that quantum spinning particles interact in this way?

Classically rotating particles, like the Earth and the Moon, generate tides, which are spin 2 effects, whereas SL(2,C) has only spin 1 force fields (electric and magnetic). The interaction group for classical rotating particles is SL(3,R), which consists of a spin 1 gravitational field and a spin 2 tidal field. So should we not rather assume that an electron and a proton behave in the same way, and that their interactions are described by SL(3,R) rather than SL(2,C)? Of course, no-one listens to this idea, since SL(2,C) has worked perfectly well since 1928, so why would anyone want to change it?

The reason we might want to change it is that it doesn’t work accurately enough for muons. Anyway, I’ve offered two alternatives, depending on your philosophy. Either SO(4) or SL(3,R). But not Spin(3,1)=SL(2,C) under any circumstances. My icosahedral model offers both at once, so that you can do quantum mechanics and relativity at the same time, in a consistent way.


11 Responses to “The mysteries of spin”

  1. Robert A. Wilson Says:

    Oh, and by the way, I nearly forgot – it is the tidal forces exerted by the electron on the proton that enable us to separate the three quarks. That is the reason why the symmetry group SL(3,R) is not only the symmetry group of quantised relativity, and (dually) of relativistic quantum mechanics, but also of the strong force that describes the internal structure of the proton (see the octions paper).

  2. brodix Says:

    Would it be more effective to describe it as cycles, rather than spin?
    Spin implies the entity is functioning in isolation, while cycles implies a connectivity/relationship/feedback loop with the context.
    Nodes and networks.
    Synchronization as centripetal dynamic pulling in, while harmonization equalizes across the field.
    Particles and fields. Organisms and ecosystems.

  3. Robert A. Wilson Says:

    Even if one assumes that the (intrinsic) spin of an electron (or a photon, for that matter) can be described without time (reasonable, perhaps, since a photon does not experience time), Woit’s assertion (it is not an argument) that spin is an infinitesimal generator of the rotation group SO(3) cannot go unchallenged. He does not define “infinitesimal”, but in mathematics it means “arbitrarily small”. But it cannot mean this in physics, because spin is quantised in units of Planck’s constant. So what does it mean? First, it means taking the tangent space, that is the Lie algebra, and then it means scaling the Lie algebra using Planck’s constant.

    But if you simply take the real Lie algebra, as Woit does, or the complex Lie algebra, as most physicists do, then there is no limit to how small the spin can actually be. This physical contradiction can only be avoided by taking an *integer* version of the Lie algebra. The integer versions of so(3) are well known, and this classification implies that elementary particles have spin at most 5/2. Under the additional assumption of isotropy of space, it also tells us that the number of distinct directions of intrinsic spin of an electron or a photon is at most 60. As I have explained many times before, this resolves the measurement problem, and cuts the Gordian knot of tortuous philosophical arguments about interpretations of quantum mechanics.

    Why does no-one listen?

    • Robert A. Wilson Says:

      I think there are, strictly speaking, just two integral forms of the Lie algebra su(2) = so(3). The classification depends on the classification of the elements of minimal norm under the Killing form. If these do not span the 3-space, then spin is a 2-dimensional concept rather than a 3-dimensional concept – I don’t necessarily rule this out, but I think most physicists would, so let’s not consider this possibility for now. There are only two other cases, geometrically, that is the cube and the hexagonal prism.

      Of these, the cube is the “obvious” one to use, but my papers indicate that the hexagonal prism is in fact the one that is supported by experiment. Strange, but true.

  4. Zsat Says:

    All this is interesting but I’m finding hard to follow your explanations about time and clocks in relation with rotation and spin. Why is the concept of time needed? And by concept of time do you include the proper time of relativity measured by ideal clocks? Also, first you say that for a clock an atom is needed, not an elementary particle, but then go on to claim an electron does tick like a clock for Euclidean rotations in four dimensions. I’m confused. Could you elaborate on why doesn’t SO(3) or SO(3,1) tick? Maybe a bit more in mathematical terms?

    • Robert A. Wilson Says:

      I guess the problem is that I have put several different arguments base on different assumptions, and it may not be clear exactly where I have changed the assumptions.

      From a mathematical point of view there are three different Lie groups one might want to use: SO(3), and/or its double cover SU(2), which has no concept of time at all; SO(3,1), and/or its double cover SL(2,C), which has a continuous time that can take arbitrary positive values; and SO(4), and/or its double cover SU(2) x SU(2), which has a discrete time that (potentially) ticks backwards and forwards between two states. The question then is, which of these groups, or their double covers, most accurately represents the actual experimental properties of an electron.

      My point of view, as a mathematician, is that of these six cases, it is SO(4) that most accurately describes the electron, at least as far as I understand the experimental properties of an electron.

    • Robert A. Wilson Says:

      There is then an entirely separate argument about which group most accurately describes the photon. The group that is always used is SO(3,1), but I argue that this is wrong for several different reasons. The most important one is that, as far as I can see, it does not accurately describe the physical properties of polarisation at the quantum level. I believe it can describe polarisation correctly in the classical wave description of light, but I do not believe it can accurately describe the quantum behaviour of entangled photons.

      For this purpose, it is necessary to accept that a photon has a direction of motion, and that its symmetry group is therefore SO(2,1) and not SO(3,1). It is necessary to accept that polarisation for an individual photon is a binary choice, not a continuous one. This fact is well supported by experiment, combined with Bell’s theorem. Now the group SO(2,1) does not provide such a binary choice, but its double cover SL(2,R) does.

      The usual assumption is that extending SO(2,1) to 3+1-dimensional spacetime requires extending to SO(3,1), and therefore extending SL(2,R) to SL(2,C). This seems “obvious”, so no-one questions it. But I question it, and I believe it is wrong. I believe what actually happens to the photon itself, *which does not experience time*, must be described by a group that *does not involve time*. This group must therefore be SL(3,R) and not SL(2,C).

  5. Zsat Says:

    Ok, thanks. I still fail to see the relation between a concept of time and spin or chirality. Mathematically time has been linked to a real parameter in dynamic systems and more recently to a fourth dimension either for classical Newtonian physics and the Galilei group or for relativistic physics and the Lorentz group. Admittedly beyond this time seems to not have been quite figured out so maybe you could share your concept of time and we then could better understand your insistence that it’s required for a theory of spin and rotation.

    • Robert A. Wilson Says:

      I don’t see how you can talk about any kind of motion without a concept of time – this includes rotations as a type of motion. Intrinsic spin could be different – I allow for that possibility – but since even intrinsic spin carries angular momentum along with it, this seems unlikely.

  6. Zsat Says:

    Sure, but since you just wrote that SO(3), a rotation kind of motion has no concept of time. Anyway, no worries, great piece, keep it up.

  7. brodix Says:

    It seems to me the problem with time isn’t that it’s complicated, but how it relates to our sentience.
    As these mobile organisms this conscious interface our body has with its situation functions as as sequence of perceptions, in order to navigate, so our experience of time is as the point of the present, moving past to future.
    While the evident reality is that activity changes future to past. Tomorrow becomes yesterday, because the earth turns.
    There is no literal dimension of time, because the past is consumed by the present, to inform and drive it. Causality and conservation of energy. Cause becomes effect.
    Energy is “conserved,” because it manifests this presence, creating time, temperature, pressure, color and sound. Time is frequency, events are amplitude.
    So energy, as this presence, goes past to future, because the patterns it generates coalesce and dissolve, rise and fall, future to past.
    Energy drives the wave, while the fluctuations rise and fall.
    Consciousness also goes past to future, while the perceptions, emotions and thoughts giving it form and structure go future to past. Though it’s the gut processing the energy, while the head sorts the information. Thus the obsession with the math, to the detriment of the physics.
    It would seem that entropy for the energy is to radiate toward infinity, while entropy for the information is to coalesce toward equilibrium.
    Thus galaxies.

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